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Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes

Léo Bigorgne, Renato Velozo Ruiz

TL;DR

This work studies the exterior of Schwarzschild for massless Vlasov fields, proving an integrated local energy decay (ILED) without relative degeneration by exploiting a trapping-weight φ_- and a carefully modified commutation vector field V_+^{mod}. A refined commutation framework, together with redshift and r^p-weighted energy methods, yields time-decay for the energy flux and, consequently, decay for the energy-momentum tensor and its first derivatives. The analysis hinges on a detailed phase-space/geometry approach to trapping near the photon sphere, a Lyapunov-type control near the horizon, and region-wise decompositions to manage degenerate derivatives. The results align with broader efforts to extend wave-equation decay techniques to kinetic equations on black hole backgrounds and are compatible with quasi-linear wave analyses on such spacetimes. Overall, the paper provides a robust toolkit for quantitative decay and regularity of massless Vlasov fields on Schwarzschild, with potential applications to nonlinear stability problems and to linear/nonlinear kinetic models in curved spacetimes.

Abstract

In this article, we make use of a weight function capturing the concentration phenomenon of unstable future-trapped causal geodesics. A projection $V_+$, on the tangent space of the null-shell, of the associated symplectic gradient turns out to enjoy good commutation properties with the massless Vlasov operator. This reflects that $V_+f$ decays exponentially locally near the photon sphere, for any smooth solution $f$ to the massless Vlasov equation. By identifying a well-chosen modification of $V_+$, we are able to construct a $W_{x,p}^{1,1}$ weighted norm for which any smooth solution to the massless Vlasov equation verifies an integrated local energy decay estimate without relative degeneration. Together with the $r^p$-weighted energy method of Dafermos--Rodnianski, we establish time decay for the energy norm. This norm allows for the control of the energy-momentum tensor $\mathrm{T}[f]$ as well as all its first order derivatives. The method developed in this paper is in particular compatible with approaches recently developed for the study of quasi-linear wave equations on black hole spacetimes.

Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes

TL;DR

This work studies the exterior of Schwarzschild for massless Vlasov fields, proving an integrated local energy decay (ILED) without relative degeneration by exploiting a trapping-weight φ_- and a carefully modified commutation vector field V_+^{mod}. A refined commutation framework, together with redshift and r^p-weighted energy methods, yields time-decay for the energy flux and, consequently, decay for the energy-momentum tensor and its first derivatives. The analysis hinges on a detailed phase-space/geometry approach to trapping near the photon sphere, a Lyapunov-type control near the horizon, and region-wise decompositions to manage degenerate derivatives. The results align with broader efforts to extend wave-equation decay techniques to kinetic equations on black hole backgrounds and are compatible with quasi-linear wave analyses on such spacetimes. Overall, the paper provides a robust toolkit for quantitative decay and regularity of massless Vlasov fields on Schwarzschild, with potential applications to nonlinear stability problems and to linear/nonlinear kinetic models in curved spacetimes.

Abstract

In this article, we make use of a weight function capturing the concentration phenomenon of unstable future-trapped causal geodesics. A projection , on the tangent space of the null-shell, of the associated symplectic gradient turns out to enjoy good commutation properties with the massless Vlasov operator. This reflects that decays exponentially locally near the photon sphere, for any smooth solution to the massless Vlasov equation. By identifying a well-chosen modification of , we are able to construct a weighted norm for which any smooth solution to the massless Vlasov equation verifies an integrated local energy decay estimate without relative degeneration. Together with the -weighted energy method of Dafermos--Rodnianski, we establish time decay for the energy norm. This norm allows for the control of the energy-momentum tensor as well as all its first order derivatives. The method developed in this paper is in particular compatible with approaches recently developed for the study of quasi-linear wave equations on black hole spacetimes.
Paper Structure (76 sections, 83 theorems, 422 equations, 1 figure)

This paper contains 76 sections, 83 theorems, 422 equations, 1 figure.

Table of Contents

  1. Introduction
  2. State of the art on decay for massless Vlasov fields on black hole exteriors
  3. Based on weighted energy methods
  4. Based on phase space volume estimates
  5. Vlasov equations on backgrounds with hyperbolic flows
  6. The main results
  7. The coordinate system induced by the tortoise coordinates
  8. Weight functions and commutation vector fields
  9. The energy fluxes
  10. Statement of the main results
  11. Difficulties of the problem
  12. Ideas and strategy of the proof
  13. Step $0$: Linearisation around the photon sphere.
  14. Step $1$: Finding a good weight function.
  15. Step $2$: Identifying a vector field commuting well with ${\mathbb{X}}_g$.
  16. ...and 61 more sections

Key Result

Theorem 1.1

Let $f$ be a sufficiently regular solution to the massless Vlasov equation vlasov_eqn_massless_intro such that $\mathcal{E}[f](0) <+\infty$. Then, there exists $C>0$, depending only on $M$, such that

Figures (1)

  • Figure 1: The foliation $(\Sigma_{\tau})_{\tau \geq 0}$.

Theorems & Definitions (193)

  • Theorem 1.1: ILED without relative degeneration
  • Corollary 1.1: Inverse polynomial decay of the energy-flux
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4: Exponential decay of the energy-flux
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 183 more